Provable Phase Retrieval with Mirror Descent

In this paper, we consider the problem of phase retrieval, which consists of recovering an n dimensional real vector from the magnitude of its m linear measurements. We propose a mirror descent (or Bregman gradient descent) algorithm based on a wisely chosen Bregman divergence, hence allowing us to remove the classical global Lipschitz continuity requirement on the gradient of the nonconvex phase retrieval objective to be minimized. We apply the mirror descent for two random measurements: the i.i.d. standard Gaussian and those obtained by multiple structured illuminations through coded diffraction patterns. For the Gaussian case, we show that when the number of measurements m is large enough, then with high probability, for almost all initializers, the algorithm recovers the original vector up to a global sign change. For both measurements, the mirror descent exhibits a local linear convergence behavior with a dimension-independent convergence rate. Finally, our theoretical results are illustrated with various numerical experiments, including an application to the reconstruction of images in precision optics.